Schwarz boundary value problem on Reuleaux triangle

Abstract

We discuss the Schwarz problem on Reuleaux triangle. By the technique of parqueting reflection principle, we translate the Schwarz problem into a Riemann boundary value problem on a system of arcs. We first state that the Riemann problem is solvable, while the solutions are represented by a Cauchy type integral. At last, as the main theorem, we prove that those two problems are equivalent to each other. And then we obtain the explicit integral representation of the solution to a Schwarz problem.